Table Of ContentElectrochemical Impedance Spectroscopy and its Applications
Andrzej Lasia
Département de chimie, Université de Sherbrooke, Sherbrooke
Québec, J1K 2R1
[email protected]
A. Lasia, Electrochemical Impedance Spectroscopy and Its Applications, Modern Aspects of
Electrochemistry, B. E. Conway, J. Bockris, and R.E. White, Edts., Kluwer Academic/Plenum
Publishers, New York, 1999, Vol. 32, p. 143-248.
Table of content
I. Introduction..............................................................................................................................4
1. Response of Electrical Circuits............................................................................................4
(i) Arbitrary Input Signal......................................................................................................4
(ii) Alternating Voltage (av) Input Signal..........................................................................6
2. Impedance of Electrical Circuits..........................................................................................7
(i) Series R-C Circuit............................................................................................................8
(ii) Parallel R-C Circuit......................................................................................................8
(iii) Series: R + Parallel R-C Circuit..................................................................................9
s
3. Interpretation of the Complex Plane and Bode Plots...........................................................9
II. Impedance measurements......................................................................................................10
1. Ac Bridges.........................................................................................................................10
2. Lissajous Curves................................................................................................................11
3. Phase Sensitive Detection (PSD).......................................................................................11
4. Frequency Response Analyzers.........................................................................................12
5. Fast Fourier Transform (FFT)............................................................................................13
(i) Pulse Perturbation..........................................................................................................14
(ii) Noise Perturbation.....................................................................................................14
(iii) Sum of Sine Waves....................................................................................................14
III. Impedance of faradaic reactions in the presence of diffusion................................................15
1. The Ideally Polarizable Electrode......................................................................................15
2. Semi-Infinite Linear Diffusion..........................................................................................15
3. Spherical Diffusion............................................................................................................20
4. Cylindrical Electrodes........................................................................................................22
5. Disk Electrodes..................................................................................................................23
6. Finite-Length Diffusion.....................................................................................................23
(i) Transmissive Boundary.................................................................................................24
(ii) Reflective Boundary..................................................................................................25
7. Analysis of Impedance Data in the Case of Semi-Infinite Diffusion: Determination of the
Kinetic Parameters.....................................................................................................................26
(i) Randles’ Analysis64,65,67.................................................................................................26
(ii) De Levie-Husovsky Analysis....................................................................................27
(iii) Analysis of cot ϕ........................................................................................................27
(iv) Sluyter's Analysis.......................................................................................................28
IV. Impedance of A faradaic reaction involving adsorption of reacting species.........................29
1. Faradaic Reaction Involving One Adsorbed Species.........................................................29
2. Impedance Plots in the Case of One Adsorbed Species....................................................31
3. Faradaic Impedance in the Case Involving Two Adsorbed Species..................................34
4. Impedance Plots in the Case of Two Adsorbed Species....................................................36
5. Faradaic Impedance for the Process Involving Three or More Adsorbed Species............36
V. Impedance of solid electrodes................................................................................................37
3
1. Frequency Dispersion and Electrode Roughness...............................................................37
2. Constant Phase Element.....................................................................................................37
3. Fractal Model.....................................................................................................................40
4. Porous Electrode Model.....................................................................................................42
(i) Porous Electrodes in the Absence of Internal Diffusion................................................42
(a) De Levie’s treatment..............................................................................................43
(b) Rigorous treatment.................................................................................................45
(ii) Porous Electrodes in the Presence of Axial Diffusion...............................................46
(iii) Other Pore Geometries...............................................................................................49
5. Generalized Warburg Element...........................................................................................49
VI. Conditions for "good" impedances........................................................................................50
1. Linearity, Causality, Stability, Finiteness..........................................................................50
2. Kramers-Kronig Transforms..............................................................................................51
3. Non-Stationary Impedances...............................................................................................53
VII. Modeling of experimental data..............................................................................................54
1. Selection of the Model.......................................................................................................54
2. CNLS Approximations......................................................................................................56
(i) CNLS Method................................................................................................................56
(ii) Statistical Weights.....................................................................................................57
(iii) AC Modeling Programs.............................................................................................58
VIII. Instrumental limitations.........................................................................................................58
IX. Conclusion.............................................................................................................................60
4
I. INTRODUCTION
Electrochemical Impedance Spectroscopy (EIS) or ac impedance methods have seen
tremendous increase in popularity in recent years. Initially applied to the determination of the
double-layer capacitance1-4 and in ac polarography,5-7 they are now applied to the
characterization of electrode processes and complex interfaces. EIS studies the system response
to the application of a periodic small amplitude ac signal. These measurements are carried out at
different ac frequencies and, thus, the name impedance spectroscopy was later adopted. Analysis
of the system response contains information about the interface, its structure and reactions taking
place there. EIS is now described in the general books on electrochemistry,8-17 specific books18,19
on EIS, and there are also numerous articles and reviews.6,20-31 It became very popular in the
research and applied chemistry. The Chemical Abstract database shows ~1,500 citations per year
of the term "impedance" since 1993 and ~1,200 in earlier years and ~500 citations per year of
"electrochemical impedance". Although the term "impedance" may include also non-
electrochemical measurements and "electrochemical impedance" may not include all the
electrochemical studies, the popularity of this technique cannot be denied.
However, EIS is a very sensitive technique and it must be used with great care. Besides,
it is not always well understood. This may be connected with the fact that existing reviews on
EIS are very often difficult to understand by non-specialists and, frequently, they do not show
the complete mathematical developments of equations connecting the impedance with the
physico-chemical parameters. It should be stressed that EIS cannot give all the answers. It is a
complementary technique and other methods must also be used to elucidate the interfacial
processes.
The purpose of this review is to fill this gap by presenting a modern and relatively
complete review of the subject of electrochemical impedance spectroscopy, containing
mathematical development of the fundamental equations.
1. Response of Electrical Circuits
(i) Arbitrary Input Signal
Application of an electrical perturbation (current, potential) to an electrical circuit causes
the appearance of a response. In this chapter, the system response to an arbitrary perturbation
and, later, to an ac signal, will be presented. Knowledge of the Laplace transform technique is
assumed, but the reader may consult numerous books on the subject.
First, let us consider application of an arbitrary (but known) potential E(t) to a resistance
R. The current i(t) is given as: i(t) = E(t)/R. When the same potential is applied to the series
connection of the resistance R and capacitance C, the total potential difference is a sum of
potential drops on each element. Taking into account that for a capacitance E(t) = Q(t) /C, where
Q is the charge stored in a capacitor, the following equation is obtained:
Q(t) 1 t (1)
E(t)= i(t)R+ = i(t)R+ ∫i(t)dt
C C
0
This equation may be solved using either Laplace transform or differentiation techniques. 32-34
Differentiation gives:
di(t) i(t) 1 dE(t) (2)
+ =
dt RC R dt
which may be solved for known E(t) using standard methods for differential equations.
5
The Laplace transform is an integral transform in which a function of time f(t) is
transformed into a new function of a parameter s called frequency, f(s) or F(s), according to:
∞
L [f(t)] = f(s) = F (s) = ∫ f(t)exp(−st)dt (3)
0
The Laplace transform is often used in solution of differential and integral equations. In general,
the parameter s may be complex, s = ν +jω, where j = −1, but in this chapter only the real
transform will be considered, i.e. s = v. Direct application of the Laplace transform to eqn. (1),
t
taking into account that L (∫i(t)dt) = i(s)/s, gives:
0
i(s)
E(s)=i(s)R + (4)
sC
which leads to:
⎛ 1 ⎞
i(s) = E(s)/⎜R+ ⎟ (5)
⎝ sC⎠
The ratio of the Laplace transforms of potential and current, E(s)/i(s) is expressed in the units of
resistance, Ω, and is called impedance, Z(s). In this case:
E(s) 1
Z(s)= = R+ (6)
i(s) sC
The inverse of impedance is called admittance, Y(s) = 1/Z(s). They are transfer functions which
transform one signal, e.g. applied voltage, into another, e.g. current. Both are called immittances.
Some other transfer functions are discussed in refs. 18, 35 and 36. It should be noticed that the
impedance of a series connection of a resistance and capacitance, eqn. (6), is a sum of the
contributions of these two elements: resistance, R, and capacitance, 1/sC.
For the series connection of a resistance, R, and inductance, L, the total potential
difference consists of the potential drop on both elements:
di(t) (7)
E(t) =i(t)R+L
dt
Taking into account that L [di(t)/dt] = s i(s) - i(0+), and taking i = 0 at t = 0, one obtains the
current response in the Laplace space:
i(s) =E(s) / (R +sL) (8)
In both cases considered above the system impedance consists of the sum of two terms,
corresponding to two elements: resistance and capacitance or inductance.
In general, one can write contributions to the total impedance corresponding to the
resistance as R, the capacitance as 1/sC and the inductance as sL. Addition of impedances is
analogous to the addition of resistances. Knowledge of the system impedance allows for an easy
solution of the problem.
For example, when a constant voltage, E , is applied at time zero to a series connection of
0
R and C, the current is described by eqn. (5). Taking into account that the Laplace transform of a
constant L (E ) = E /s, one gets:
0 0
E E 1
i(s) = 0 = 0 (9)
s(R +1/ sC) R s+1/ RC
Inverse transform of (9) gives the current relaxation versus time:
6
E
i(t) = 0 exp(−t / RC) (10)
R
The result obtained shows that after the application of the potential step, current initially equals
E /R and it decreases to zero as the capacitance is charged to the potential difference E .
0 0
Similarly, application of the potential step to a series connection of R and L produces
response given by eqn. (8) which, after substitution of E(s) = E /s, gives:
0
E E 1 E ⎛1 1 ⎞
i(s) = 0 = 0 = 0 + (11)
⎜ ⎟
s(R +sL) L s(s+ R / L) R ⎝s s+ R / L⎠
Inverse transform gives the time dependence of the current:
E ⎡ ⎛ Rt⎞⎤
i(t) = 0 1−exp − (12)
⎢ ⎜ ⎟⎥
R ⎣ ⎝ L ⎠⎦
The current starts at zero as the inductance constitutes infinite resistance at t = 0 and it increases
to E /R as the effect of inductance becomes negligible in the steady-state condition.
0
In a similar way other problems of transient system response may be solved. More
complex examples are presented, e.g., in refs. 33-34. It should be added that an arbitrary signal
may be applied to the system and if the Laplace transforms of the potential and current are
determined, e.g. by numerical transform calculations, the system impedance is determined. In the
Laplace space the equations (e.g. eqns. (9) and (11)) are much simpler than those in the time
space (e.g. eqns. (10) and (12)) and analysis in the frequency space s allows for the
determination of the system parameters. This analysis is especially important when an ideal
potential step cannot be applied to the system because of the band-width limitations of the
potentiostat.37 In this case it is sufficient to know i(t) and the real value of the potential applied to
the electrodes by the potentiostat, E(t), which allows numerical Laplace transformation to be
carried out and the system impedance obtained.
In the cases involving more time constants, i.e. more than one capacitance or inductance
in the circuit, the differential equations describing the system are of the second or higher order
and the impedances obtained are the second or higher order functions of s.
(ii) Alternating Voltage (av) Input Signal
In the EIS we are interested in the system response to the application of a sinusoidal
signal, e.g.: E = E sin(ωt), where E is the signal amplitude, ω = 2πf is the angular frequency,
0 0
and f is the av signal frequency. This problem may be solved in different ways. First, let us
consider application of an av signal to a series R-C connection. Taking into account that the
Laplace transform of the sine function L [sin(ωt)] = ω/(s2 + ω2)], use of eqn. (5) gives:
E ω 1 E ω 1 1
i(s) = 0 = 0 (13)
s2 +ω2 R +1/ sC R s2 +ω2 s+1/ RC
Distribution into simple fractions leads to:
E0 ⎡ 2 ω ω s ω 1 ⎤ (14)
i(s)= ⎢ω + − ⎥
R[ω2 +(1/ RC)2]⎣ s2 +ω2 RC s2 +ω2 RC s+1/ RC⎦
and the inverse Laplace transform, taking L -1 [s/(s2+ω2) = cos ωt, gives:
E0 ⎡ 2 ω ω ⎤ (15)
i(t)= ω sin(ωt)+ cos(ωt)− exp(−t/ RC)
R[ω2 +(1/ RC)2]⎣⎢ RC RC ⎦⎥
7
The third term in eqn. (15) corresponds to a transitory response observed just after application of
the av signal and it decreases quickly to zero. The steady-state equation may be rearranged into a
simpler form:
i(t)= E0 ⎡sin(ωt)+ 1 cos(ωt)⎤ (16)
⎢ ⎥
⎛ 1 ⎞ ⎣ ωRC ⎦
R⎜1+ ⎟
⎝⎜ (ωRC)2⎠⎟
and by introducing tan ϕ = 1/ωRC the following form is found:
E E
i(t)= 0 sin(ωt+ϕ)= 0 sin(ωt+ϕ) (17)
1 |Z|
R2 +
(ωC)2
where ϕ is the phase-angle between current and potential, ϕ = arctan(1/ωRC). It is obvious that
the current has the same frequency as the applied potential but is phase-shifted by the angle ϕ.
The value |Z| has units of resistance; it is the length of a vector obtained by addition of two
perpendicular vectors: R and 1/ωC.
2. Impedance of Electrical Circuits
In order to simplify the calculations of impedances, the result obtained for the periodic
perturbation of an electrical circuit may be represented using complex notation. In the latter
example the system impedance, Z(jω), may be represented as:
1 1 (18)
(cid:3)
Z(jω)≡ Z = Z'+jZ''= R+ = R− j
jωC ωC
and the real and imaginary parts of the impedance are: Z’ = R and Z” = -1/ωC, respectively. It
should be noted that the complex impedance Z(jω), eqn. (18), may be obtained from Z(s), eqn.
(6), by substitution: s = jω. In fact, this is the imaginary Laplace transform. The modulus of
Z(jω), eqn. (17), equals:
|Z| = (Z')2 +(Z")2 = R2 +(1/ωC)2 (19)
^
and the phase-angle between the imaginary and real impedance equals ϕ ≡ arg(Z) =
atan(-1/ωRC). It should be noticed that the sign of ϕ, between potential and current, described
above for the impedances, is different from that found between current and potential, eqn. (17). It
may be recalled that in complex notation:
Z(jω)=|Z|exp(jϕ)=|Z|[cos(ϕ)+ jsin(ϕ)] (20)
Analysis of eqn. (17) indicates that the current represents a vector of the length i = E /|Z| which
0 0
rotates with the frequency ω. Current and potential are rotating vectors in the time domain, as
represented in Figure 1a. Using complex notation they may be described by:
E = E exp(jωt) and i =i exp[j(ωt+ϕ)]
0 0 (21)
These vectors rotate with a constant frequency ω and the phase-angle, ϕ, between them stays
constant. Instead of showing rotating vectors in time space it is possible to present immobile
Figure 1.
8
vectors in the frequency space, separated by the phase-angle ϕ. These vectors are called phasors;
~ ~
they are equal to E = E and I = I exp(jϕ), where the initial phase shift of the potential was
0 0
assumed to be zero, see Figure 1b.
In general, the complex impedance may be written for any circuit by taking R for a
resistance, 1/jωC for a capacitance and jωL for an inductance, and applying Ohm’s and
Kirchhoff’s laws to the connection of these elements. Several examples of this method are
presented below.
(i) Series R-C Circuit
In the case of a series connection of the resistance and capacitance the impedance is
given by: Z(jω) = R + 1/jωC = R - j/ωC. The result may be represented graphically using two
types of plots: complex plane (also known as Argand or Nyquist plots) and Bode plots. The
complex plane plot is a plot of Z” versus Z’, that is, the imaginary versus the real components,
plotted for various frequencies. A complex plane plot for a series connection R-C (R = 100 Ω, C
= 2×10-5 F) circuit is shown in Figure 2. It consists of a straight line perpendicular to the real
axis. Other types of graphs are Bode plots i.e. log |Z| (magnitude) and phase-angle, ϕ, versus log
ω. They are also shown in Figure 2. The graph of log |Z| versus log ω, Figure 2d, contains one
breakpoint or corner frequency. This point corresponds to the system characteristic frequency ω
= 1/RC = 500 s-1 or a time constant τ = RC = 0.002 s. The phase-angle changes from 90o at low
frequencies to 0 at high frequencies. This circuit corresponds to an ideally polarized electrode in
solution, e.g. a mercury electrode - supporting electrolyte solution.
Figure 2.
The complex plane plots may also be obtained for admittances. Admittance for the series
R-C connection equals:
1 1 R j
Y(jω)= = = +
Z(jω) R− j R2 + 1 ωC⎛⎜R2 + 1 ⎞⎟ (22)
ωC ω2C2 ⎝ ω2C2⎠
It represents a semi-circle on the complex plane plot, Figure 2c. It should be stressed that for
capacitive circuits the imaginary impedance is always negative and the imaginary admittance is
positive.
(ii) Parallel R-C Circuit
For the parallel R-C connection the total admittance equals: Y(jω)=1/R+ jωC such
that:
2 (23)
1 R R jωR C
Z(jω)= = = −
1/ R+ jωC 1+ jωRC 1+ω2R2C2 1+ω2R2C2
ˆ ˆ
There are two limits of the impedance: ω = 0, Z = R and ω → ∞, Z = 0. The corresponding
complex plane and Bode plots, for the same values of R and C elements as used in the series R-C
model above, are shown in Figure 3. The Nyquist plot shows a semicircle of radius R/2 and the
center on the real axis, and the frequency at the semicircle maximum equal to: ω = 1/RC. The
circuit’s characteristic breakpoint frequency (inverse of the characteristic time constant), as
observed in the impedance Bode graph, is the same as for the series and the parallel R-C circuits.
9
The complex plane admittance plot represents a straight line parallel to the imaginary axis,
Figure 3c, which is similar to the impedance complex plane plot for the series R-C connection.
Figure 3.
(iii) Series: R + Parallel R-C Circuit
s
Finally, impedance of the circuit shown in Figure 4, consisting of a series connection of
the resistance R with the parallel connection of R -C , is given as:
s ct dl
Figure 4.
1 (24)
Z(jω)= R +
s
1/ R + jωC
ct dl
The corresponding complex plane and Bode plots are also shown in Figure 4 for R = 100 Ω, R
ct s
= 10 Ω and C = 20 µF. The main difference between circuits in Figure 3 and Figure 4 is
dl
connected with the fact that in the latter circuit, at ω → ∞, Z → R and ϕ → 0, due to the
s
presence of R , and for ω → 0 Z → R + R . The frequency corresponding to the maximum of
s s ct
Z’’ is still equal to ω = 1/R C = 500 s-1. In addition, the Bode log |Z| plot shows that there are
ct dl
two breakpoints (bends). For comparison, the admittance complex plane plot is also shown in
Figure 4c.
3. Interpretation of the Complex Plane and Bode Plots
Complex plane (Nyquist) plots are the most often used in the electrochemical literature
because they allow for an easy prediction of the circuit elements. However, they do not show all
details; for example, exactly the same Nyquist impedance plots, shown in Figure 3 and Figure 4,
may be obtained for different values of the capacitance C. The only difference between them will
be the fact that the points on the semicircle would correspond to different frequencies.
Nevertheless, Nyquist plots allow for an easy relation to the electrical model. On the other hand
Bode plots contain all the necessary information. That is why Bode plots are mainly used in the
circuit analysis. The Bode magnitude plots may be easily predicted from the circuit impedance.33
Let us consider the circuit shown in Figure 4a. Its impedance is presented by eqn. (24). This
equation may be rearranged into another form:
⎛ R R C ⎞
s ct dl (25)
1+ jω⎜ ⎟
(cid:3) ⎝ Rs + Rct ⎠ 1+ jωτ2
Z =(R + R ) =(R + R )
s ct ( ) s ct
1+ jω R C 1+ jωτ
ct dl 1
where τ and τ are the Bode characteristic time constants. From eqn. (25) log(|Z|) is easily
1 2
evaluated:
log(|Z|)=log(|R + R |)+log(|1+ jωτ |)−log(|1+ jωτ |)
s ct 2 1 (26)
In order to construct asymptotic lines in the Bode magnitude plot, the contribution of each term
in eqn. (26) can be considered independently and then their sum may be easily obtained. Each
term log(|1+jωτ|) has two limits: when ωτ<<1, i.e. ω<<1/τ, log(|1+jωτ|) = 0 and when ωτ>>1,
10
log(|1+jωτ|) = log τ +log ω, which correspond to a straight line with a slope of one and intercept
log ω = -log τ.
The graphs corresponding to these lines are shown in Figure 5. The break-point
frequencies in the Bode magnitude plot, Figure 4d and Figure 5, are ω = 1/τ = 500 s-1 and ω =
1 1 2
1/τ = 5500 s-1. The continuous line is the sum of the three asymptotes. In this way Bode
2
magnitude graphs may be constructed for other circuits.
The Bode phase-angle graph is shown in Figure 4e. The phase-angle is described by:
⎛ 2 ⎞
ωR C
⎜ ct dl ⎟
ϕ=atan(Z''/Z')=atan⎜ ⎟
2
⎜ R + R +(ωR C ) R ⎟ (27)
⎝ s ct ct dl s⎠
It can be shown that this function has a maximum at:
1 RS + Rct (28)
ω =
R C R
ct dl s
which, in this case, equals ω = 1658 s-1. It should be noticed that the maximum of the phase-
angle is different from the maximum of the imaginary part of the impedance, corresponding to
the maximum of the semicircle at Z’ = R + R /2 at ω = 1/R C . The plots of Z' and Z'' (or their
s ct ct dl
logarithms) as a function of log ω are also sometimes shown in the literature.
Figure 5.
II. IMPEDANCE MEASUREMENTS
Dc transient response of electrochemical systems is usually measured using potentiostats.
In the case of EIS an additional perturbation is added to the dc signal in order to obtain the
frequency response of the system. The system impedance may be measured using various
techniques:
1) ac bridges
2) Lissajous curves
3) phase sensitive detection (PSD)
4) frequency response analysis (FRA)
5) fast Fourier transform (FFT)
Because older techniques were described in detail in refs. 18, 19, 26, 28, 30 and 31, this chapter
will be focused on the last three techniques.
1. Ac Bridges
This technique was the first used to measure the double-layer parameters (principally of
the dropping mercury electrode) and, later, to measure the electrode impedance in the presence
of a faradaic reaction to determine the kinetics of electrode processes. The use of ac bridges
provides a very good precision of measurements. It has been described in detail in refs. 18, 26,
28 and 38. The ac bridge with potentiostatic control may also be used. Although this method is
slow, because bridge compensation must be carried out at each frequency manually, it is still
used, principally in precise double-layer measurements.39-41
Description:Electrochemical Impedance Spectroscopy and its Applications. Andrzej Lasia. Département de chimie, Université de Sherbrooke, Sherbrooke.
